While the J-Integral was developed for nonlinear elastic material behavior, it has been extensively studies for its direct application to describing elastic-plastic material behavior [Begley & Landes, 1972; Verette & Wilhem, et al., 1973; Landes, et al., 1979; Paris, 1980; Roberts, 1981].  Its nonlinear elastic foundation has provided engineers with some techniques which allow them to focus on the combination of linear-elastic and plastic-strain hardening behavior and then to separate these two components for further study of the plastic behavior.  The J-Integral for an elastic-plastic material is taken as the sum of two components parts: the linear elastic part (J_el) and the plastic-strain hardening part (J_pl), i.e.,

which when used in conjunction with Equation 2.2.36 becomes

Engineering estimates of J then focus on the development of the plastic-strain hardening part J_pl.  Recently, Shih and coworkers have published a series of reports and technical papers [Shih & Kumar, 1979; Kumar, et al., 1980; Shih, 1976; Kumar, et al., 1981] detailing how the J_pl term can be calculated from a series of finite element models that consider changes in material properties for the same structural geometry.  The following briefly describes the Shih and coworkers method for estimating J_pl.

First, the material is assumed to behave according to a power hardening constitutive (s - e) law of the form

where a is a dimensionless constant, s_o = Ee_o, and n is the hardening exponent.  For n = 1, the material behaves as a linearly elastic material; as n approaches infinity, the material behaves ore and more like a perfectly plastic material.  For a generalization of Equation 2.2.45 to multiaxial states via the J₂ deformation theory of plasticity, Ilyushin [1946] showed that the stress at each point in the body varies linearly with a single load such as s, the remotely applied stress, under certain conditions.

Ilyushin’s analysis allowed Shih and Hutchison [1976] to use the relationship for crack tip stresses under contained plasticity, i.e. to use [Hutchinson, 1968; Rice & Rosengren, 1968]

and similar equations for s_yy, s_xy, etc., to relate the crack tip parameters uniquely to the remotely applied load.  Note that J_pl term in Equation 2.2.46 acts as a (plastic) stress field magnification factor similar to that of the stress-intensity factor in the elastic case.  The form of the relationship that Shih and Hutchinson postulated is given by

where  is a function only of relative width (a/b) and n.  An alternate form of Equation 2.2.47 that has been previously used in computer codes [Kumar, et al., 1980; Kumar, et al., 1981; Weerasooriya & Gallagher, 1981] is

where P is the applied load (per unit thickness), P^T_o is the theoretical limit load (per unit thickness), f₁ is a function only of geometry and crack length, while h₁ depends on geometry, crack length, and the strain hardening exponent n. Shih and coworkers [Kumar, et al., 1980; Kumar, et al., 1981] have tabulated the functions for a number of geometries for conditions of plant stress and plane strain.  From the reference tabulated data [also see Weerasooriya & Gallagher, 1981], these functions can be obtained by interpolation for any value within the a/b and n limits given; thus, the plastic (strain hardening) component of Equation 1.3.44 can be computed for any given applied load P from Equation 2.2.48.